The tourists are of "Chinese origin," according to a statement from the Kruger National Park.
One of the tourists said he was from Hong Kong, according to a hospital official.
An elephant in the park attacked the vehicle on a road at 6.30am on Monday, and a medical team in a helicopter rushed to help the injured male driver, said the statement.
He was taken to Clinix Phalaborwa Private Hospital, near the park.
An official at the hospital said the man was in critical condition with multiple rib fractures, and that his female companion was also being treated there for a pelvis fracture.
He said officials planned to transfer the pair to a hospital in the Pretoria area. The official spoke on condition of anonymity because he was not authorised to speak to the media.
It is unclear why the elephant became aggressive, said the park spokesperson William Mabasa. He is appealing to the public to be alert in Kruger park and try not to get too close if they see an elephant approaching on the road.
The vast Kruger park lies in South Africa's northeast, next to Zimbabwe and Mozambique. – Sapa-AP